Untitled

Abstract

We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. We show that a similar result is false for the root lattice E 8 . For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of E 8 . The new lattice yields a sphere covering which is more than 12% less dense than the formerly best known given by the lattice A * 8 . Currently, the Leech lattice is the first and only known example of a locally optimal lattice covering having a non-simplicial Delone subdivision. We hereby in particular answer a question of Dickson posed in 1968. By showing that the Leech lattice is rigid our answer is even strongest possible in a sense.

Random Sequential Covering

“…etc. Generally for L ≥ 3, the probability q L satisfies the recurrence (10) which is established similarly to the recurrence (7). By inserting (8) into (10) we find that q L stabilizes for L ≥ 3:…”

“…If (k, k + 1) with 2 ≤ k ≤ L is the first deposited dimer, (1, 2) will be the left-most dimer with probability p k−1 . This explains the sum on the right-hand side of (7). The recurrence (7) admits a simple solution:…”

“…Indeed, (1, 2) may be the first deposited dimer. This event occurs with probability (L + 1) −1 and explains the first term on the right-hand side of (7). If (k, k + 1) with 2 ≤ k ≤ L is the first deposited dimer, (1, 2) will be the left-most dimer with probability p k−1 .…”

“…A natural guess is that the least dense sphere covering can be obtained by enlarging the radii of the densest sphere packing. This is true [5,6] in two dimensions but false in three and eight dimensions [7,8], and probably false in all d ≥ 3 with the possible exception of d = 24. Optimal, that is, least dense, sphere coverings are more complicated than the densest sphere packings.…”

Periodic Triangulations of $\mathbb{Z}^n$

“…Note that existence of a periodic centrally symmetric non-Delaunay triangulation for n = 8 was established in [18].…”